Saad Bin NasirECE-6133 Final Project

EIGER (EIG ALGORITHM IMPLEMENTATION)

THE ALGORITHM

Input = netlist

Adjacency (A) and degree (D) matrices are computed

Get laplacian (L) = A-D matrix

Eigenvector corresponding to 2nd smallest eigenvalue of L gives a 1-dimensional placement of nodes

ratio cut heuristic is computed within area skew constraint

Output = partition corresponding to minimum ratio cut

IMPLEMENTATION Matrices => Matlab should be the first choice

Use power of matlab Vectorization – minimize # of loops required Pre-allocation – saves huge amounts of runtime

Making “A” and “D” matrices from netlist takes time

Simplest model (V=149 nodes) took no time!

10000 node netlist never finished!

It turns out Matrices are big!

*From Cody Planteen’s slide

A closer look reveals!

Special structures in Matrices (Toeplitz, Hermition, Symmetric,etc)

Dimensional Analysis! Is it low rank? Seemingly, yes Positive definite? No but Eigenvalues ≥ 0 so it is semi positive

definite

Sparse and Symmetric

Implementation Sparse Algebra store only non-zero enteries, rest are assumed 0 by default Three vector required for one matrix row vector, column vector and the value vector

Trade offs saves memory algorithm translation required could be slow

matrix operations require more time specialized eigenvalue calculation engine required

Storage Requirement Comparison

0

200

400

600

800

1000

1200

1400

1600

1800

2000

fract p2 industry3

linear

sparse

173 KB31KB 71 MB

1.3 MB

1810 MB

240MB

Storage gain = f (sparsity) = f (# of non-zero enteries)

Implementation Trade-off Use linear algebra for speed

Use sparse algebra for space

mixed approach gives the required balance Trade-off memory for precision laplacian matrix requires large memory Perform sparse computations till eigenvector calculation Precision needed for eigenvalue/eigenvector calculation

(Matlab uses ARPACK) Perform ratio cut heuristics on full laplacian matrix (stored in

lower precision, 32-bit floating point)

1-Dimensional placement

Validity : ∑ (placement value)2 =1

ratio cut/cut size plot

Industry3.hgr Total nodes = 15406

Results!

Design Total Cells

LeftPartition

RightPartition

Area Skew(%)

RunTime(s)

Cut size Ratio Cut

fract 149 75 74 0.67 0.42 23.75 4.3e-3

p1 902 455 447 0.88 0.569 84.7857 4.16e-4

structP 1952 985 967 0.92 1.19 174.25 1.83e-4

p2 3029 1529 1500 0.95 24.34 313.29 1.36e-4

biomedP 6514 3289 3225 0.98 49.46 1.85e+3 1.74e-4

industry2 13419 6709 6710 0.0074 61.85 720.5 1.6e-5

industry3 15406 7780 7626 0.99 78.64 1.86e+3 3.14e-4

ibm01 14111 7125 6986 0.98 43.85 380.6 7.64e-6

ComparisonDesign Runtime(s) (this work) Runtime(s) (Cody’s Work)

fract 0.42 0

p1 0.569 0

structP 1.19 8

p2 24.34 28

biomedP 49.46 279

Industry2 61.85 2094

industry3 78.64 -

ibm01 43.85 2153

For smaller designs - linear matrix computations are fastFor larger designs – using power of sparsity pays dividends

Conclusion and Extensions Significant gains in runtime and storage requirements

achieved using a combination of linear and sparse algebra

Optimized package can be written in a language like C

Gives motivation and insights in to exploring the usage of sparse algebra based mathematics into solving the problems of physical design of VLSI

Exciting mathematics being developed for low rank matrices Randomization, compressed sensing, etc

Questions?

Thank you